The Classification of Two-Dimensional Extended Topological Field Theories

Ph.D. Dissertation (UC Berkeley) 2009

 

This work proves the cobordism hypothesis in dimension two. Specifically, this work classifies 2-dimensional extended topological field theories in terms of generators and relations. In this context, a topological field theory is a functor from a bordism category to some target category and an extended field theory is a higher categorical version of this. Why use higher categories? They allow you to mathematically encode the locality of the theory, something which is predicted from the physical point of view and which is very helpful mathematically.

The methods combined algebraic results on symmetric monoidal bicategories with a generalization of Cerf theory. This provides and alternate approach to the cobordism hypothesis to the one given by Hopkins and Lurie, and to date (Feb 2012) remains the only complete account.

abstract

We provide a complete generators and relations presentation of the 2-dimensional extended unoriented and oriented bordism bicategories as symmetric monoidal bicategories. Thereby we classify these types of 2-dimensional extended topological field theories with arbitrary target bicategory. As an immediate corollary we obtain a concrete classification when the target is the symmetric monoidal bicategory of algebras, bimodules, and intertwiners over a fixed commutative ground ring. In the oriented case, such an extended topological field theory is equivalent to specifying a (non-commutative) separable symmetric Frobenius algebra.

We review the notion of symmetric monoidal bicategory, giving also a precise notion of generators and relations in this context. We provide several supporting lemmas, one of which provides a simple list of criteria for determining when a morphism of symmetric monoidal bicategories is an equivalence. We introduce the symmetric monoidal bicategory of bordisms with structure, where the allowed structures have a suitable sheaf or stack gluing property.

We modify standard techniques of Cerf theory to obtain a bicategorical decomposition theorem for surfaces. Moreover these techniques produce a finite list of local relations which are sufficient to pass between any two decompositions. We deliberately avoid the use of the classification of surfaces, and consequently our techniques are readily adaptable to higher dimensions. Although constructed for the unoriented case, our decomposition theorem is engineered to generalize to the case of bordisms with structure. We demonstrate this for the case of bordisms with orientations, which leads to a similar classification theorem.

This is the author’s 2009 Ph. D. dissertation, with minor alterations.

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One response to “The Classification of Two-Dimensional Extended Topological Field Theories

  1. Pingback: Reconstructing 4-manifolds from Morse 2-functions | Chris Schommer-Pries

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